Related papers: Properly embedded and immersed minimal surfaces in…

In this paper, on the first, we prove $\Delta r=2H$ where $\Delta $ is the Laplacian operator, $r=\left( r_{1},r_{2},r_{3}\right) $ the position vector field and $H$ is the mean curvature vector field of a surface $\mathcal{S}$ in the…

For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ is positive and proper. Furthermore,…

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal…

In this paper, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into $L_1$. As a corollary, we obtain that all planar graphs which are 1-skeletons of planar non-positively…

We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a…

We prove that a (branched) minimal immersion from $\mathbb{C}$ to $\mathbb{R}^n$ is stable if and only if it lives in an even dimensional affine subspace and is holomorphic for some orthogonal complex structure on the subspace. More…

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether…

Given $a,b\in\mathbb{R}$ and $\Phi\in C^1(\mathbb{S}^2)$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb{R}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi(N)$, where…

We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least…

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$ which…

We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of…

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of…

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral…

This paper is the first in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such…

In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus…

Let $M\subset\mathbb{R}^3$ be a properly embedded, connected, complete surface with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively -…

We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem…

One of the oldest open problems in the classical function theory is whether every open Riemann surface admits a proper holomorphic embedding into C^2. In this paper we prove the following Theorem: If D is a bordered Riemann surface whose…

In this article we consider surfaces in the product space $\h^2\times \r$ of the hyperbolic plane $\h^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete…